It is common for these notations to be written as \(f = \mathcal{O}(g)\), however this is confusing as it then seems as if the notation is symmetric or \(f\) is a function of \(g\). We prefer to use \(f \in \mathcal{O}(g)\) instead.
An overview of the most common Bachmann-Landau symbols: \[ \begin{align} f \in o(g) \iff& \forall k > 0 : \exists n_0 : \forall n > n_0 : \abs{f(n)} < k \cdot g(n) \\ f \in \mathcal{O}(g) \iff& \exists k > 0 : \exists n_0 : \forall n > n_0 : \abs{f(n)} \leq k \cdot g(n) \\ f \in \Theta(g) \iff& \exists k_1, k_2 > 0 : \exists n_0 : \forall n > n_0 : k_1 \cdot g(n) \leq f(n) \leq k_2 \cdot g(n) \\ f \sim g \iff& \lim_{x \to +\infty} \frac{f(x)}{g(x)} = 1 \\ f \in \Omega(g) \iff& \exists k > 0 : \exists n_0 : \forall n > n_0 : f(n) \geq k \cdot g(n) \\ f \in \omega(g) \iff& \forall k > 0 : \exists n_0 : \forall n > n_0 : \abs{f(n)} > k \cdot \abs{g(n)} \end{align} \]
NOTE: we give \(\Omega\) as used in complexity theory, in number theory another, weaker, definition is common.
A couple of idenities: \[ \begin{align} f \in \Omega(g) \iff& g \in \mathcal{O}(f) \\ f \in \Theta(g) \iff& f \in \mathcal{O}(g) \land f \in \Omega(g) \\ f \sim g \iff& g \sim f \\ \end{align} \]
Remember, Quicksort has average time complexity \(T(n) \sim 1.39 \, n \log_2 n\)